Are All Two-Dimensional Manifolds Conformally Flat?

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Why all two dimensional manifolds are conformally flat?
Why all manifolds with constant sectional curvature are conformally flat?
Does anyone know proofs of above statements.
Thanks in advance.
 
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paweld said:
Why all two dimensional manifolds are conformally flat?
Why all manifolds with constant sectional curvature are conformally flat?
Does anyone know proofs of above statements.
Thanks in advance.

On a surface one can always find coordinates where the metric is a scalar times the standard flat metric on the coordinate tangent vectors. You should try to prove this yourself. It was first proved by Gauss. You will get a PDE.

For constant sectional curvature I will have to think about it.
 

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